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Probabilistic Methods · Variational Inference

Joohwan Ko, Kyurae Kim, Woo Chang Kim, Jacob Gardner

Variational families with full-rank covariance approximations are known not to work well in black-box variational inference (BBVI), both empirically and theoretically. In fact, recent computational complexity results for BBVI have established that full-rank variational families scale poorly with the dimensionality of the problem compared to *e.g.* mean-field families. This is particularly critical to hierarchical Bayesian models with local variables; their dimensionality increases with the size of the datasets. Consequently, one gets an iteration complexity with an explicit $\mathcal{O}(N^2)$ dependence on the dataset size $N$. In this paper, we explore a theoretical middle ground *between* mean-field variational families and full-rank families: *structured* variational families. We rigorously prove that certain scale matrix structures can achieve a better iteration complexity of $\mathcal{O}\left(N\right)$, implying better scaling with respect to $N$. We empirically verify our theoretical results on large-scale hierarchical models.

Probabilistic Methods · Gaussian Processes

Tianjun Ke, Haoqun Cao, Feng Zhou

Bayesian few-shot classification has been a focal point in the field of few-shot learning. This paper seamlessly integrates mirror descent-based variational inference into Gaussian process-based few-shot classification, addressing the challenge of non-conjugate inference. By leveraging non-Euclidean geometry, mirror descent achieves accelerated convergence by providing the steepest descent direction along the corresponding manifold. It also exhibits the parameterization invariance property concerning the variational distribution. Experimental results demonstrate competitive classification accuracy, improved uncertainty quantification, and faster convergence compared to baseline models. Additionally, we investigate the impact of hyperparameters and components. Code is publicly available at https://github.com/keanson/MD-BSFC.

Probabilistic Methods · Variational Inference

Tom Huix, Anna Korba, Alain Oliviero Durmus, Eric Moulines

Variational inference (VI) is a popular approach in Bayesian inference, that looks for the best approximation of the posterior distribution within a parametric family, minimizing a loss that is (typically) the reverse Kullback-Leibler (KL) divergence. Despite its empirical success, the theoretical properties of VI have only recently received attention, and is restricted to the Gaussian case. This research paper aims to contribute to the theoretical study of VI in the non-Gaussian case by investigating the setting of Mixture of Gaussians with fixed covariance. In this view, VI over this specific family can be casted as the minimization of a Mollified relative entropy, i.e. the KL between the convolution (with respect to a Gaussian kernel) of an atomic measure supported on Diracs, where the support of the atomic measure correspond to the localization of the Gaussian components, and the target distribution. Hence, solving variational inference is equivalent to optimizing the positions of the Diracs (the particles), which can be done through gradient descent and takes the form of an interacting particle system. We study two sources of error in variational inference in this context. The first is an optimization result that is a descent lemma establishing that the algorithm decreases the objective at each iteration. The second is an approximation error that upper bounds the mollified relative entropy between an optimal finite mixture and the target distribution.

Reinforcement Learning

Xiaoyan Hu, Farzan Farnia, Ho-fung Leung

Reinforcement learning (RL) problems where the learner attempts to infer an unobserved reward from some feedback variables have been studied in several recent papers. The setting of Interaction-Grounded Learning (IGL) is an example of such feedback-based reinforcement learning tasks where the learner optimizes the return by inferring latent binary rewards from the interaction with the environment. In the IGL setting, a relevant assumption used in the RL literature is that the feedback variable $Y$ is conditionally independent of the context-action $(X,A)$ given the latent reward $R$. In this work, we propose *Variational Information-based IGL (VI-IGL)* as an information-theoretic method to enforce the conditional independence assumption in the IGL-based RL problem. The VI-IGL framework learns a reward decoder using an information-based objective based on the conditional mutual information (MI) between the context-action $(X,A)$ and the feedback variable $Y$ observed from the environment. To estimate and optimize the information-based terms for the continuous random variables in the RL problem, VI-IGL leverages the variational representation of mutual information and results in a min-max optimization problem. Theoretical analysis shows that the optimization problem can be sample-efficiently solved. Furthermore, we extend the VI-IGL framework to general $f$-Information measures in the information theory literature, leading to the generalized $f$-VI-IGL framework to address the RL problem under the IGL condition. Finally, the empirical results on several reinforcement learning settings indicate an improved performance in comparison to the previous IGL-based RL algorithm.

Probabilistic Methods

Liam Hodgson, Danilo Bzdok

The multivariate hypergeometric distribution describes sampling without replacement from a discrete population of elements divided into multiple categories. Addressing a gap in the literature, we tackle the challenge of estimating discrete distributions when both the total population size and the category sizes are unknown. Here, we propose a novel solution using the hypergeometric likelihood to solve this estimation problem, even in the presence of severe under-sampling. Our approach accounts for a data generating process where the ground-truth is a mixture of distributions conditional on a continuous latent variable, as seen in collaborative filtering, using the variational autoencoder framework. Empirical data simulation demonstrates that our method outperforms other likelihood functions used to model count data, both in terms of accuracy of population size estimate and learning an informative latent space. We showcase our method's versatility through applications in NLP, by inferring and estimating the complexity of latent vocabularies in reading passage excerpts, and in biology, by accurately recovering the true number of gene transcripts from sparse single-cell genomics data.

Reinforcement Learning

Haotian Fu, Pratyusha Sharma, Elias Stengel-Eskin, George Konidaris, Nicolas Le Roux, Marc-Alexandre Côté, Xingdi Yuan

We present an algorithm for skill discovery from expert demonstrations. The algorithm first utilizes Large Language Models (LLMs) to propose an initial segmentation of the trajectories. Following that, a hierarchical variational inference framework incorporates the LLM-generated segmentation information to discover reusable skills by merging trajectory segments. To further control the trade-off between compression and reusability, we introduce a novel auxiliary objective based on the Minimum Description Length principle that helps guide this skill discovery process. Our results demonstrate that agents equipped with our method are able to discover skills that help accelerate learning and outperform baseline skill learning approaches on new long-horizon tasks in BabyAI, a grid world navigation environment, as well as ALFRED, a household simulation environment.

Probabilistic Methods · Bayesian Models and Methods

Gerardo Duran-Martin, Matias Altamirano, Alex Shestopaloff, Leandro Sánchez-Betancourt, Jeremias Knoblauch, Matt Jones, Francois-Xavier Briol, Kevin Murphy

We derive a novel, provably robust, efficient, and closed-form Bayesian update rule for online filtering in state-space models in the presence of outliers and misspecified measurement models. Our method combines generalised Bayesian inference with filtering methods such as the extended and ensemble Kalman filter. We use the former to show robustness and the latter to ensure computational efficiency in the case of nonlinear models. Our method matches or outperforms other robust filtering methods (such as those based on variational Bayes) at a much lower computational cost. We show this empirically on a range of filtering problems with outlier measurements, such as object tracking, state estimation in high-dimensional chaotic systems, and online learning of neural networks.

Pierre Clavier, Tom Huix, Alain Oliviero Durmus

In this paper, we introduce and analyze a variant of the Thompson sampling (TS) algorithm for contextual bandits. At each round, traditional TS requires samples from the current posterior distribution, which is usually intractable. To circumvent this issue, approximate inference techniques can be used and provide samples with distribution close to the posteriors. However, current approximate techniques yield to either poor estimation (Laplace approximation) or can be computationally expensive (MCMC methods, Ensemble sampling...). In this paper, we propose a new algorithm, Varational Inference TS $\mathtt{VITS}$, based on Gaussian Variational Inference. This scheme provides powerful posterior approximations which are easy to sample from, and is computationally efficient, making it an ideal choice for TS. In addition, we show that $\mathtt{VITS}$ achieves a sub-linear regret bound of the same order in the dimension and number of round as traditional TS for linear contextual bandit. Finally, we demonstrate experimentally the effectiveness of $\mathtt{VITS}$ on both synthetic and real world datasets

Probabilistic Methods · Variational Inference

Ziheng Cheng, Longlin Yu, Tianyu Xie, Shiyue Zhang, Cheng Zhang

Semi-implicit variational inference (SIVI) extends traditional variational families with semi-implicit distributions defined in a hierarchical manner. Due to the intractable densities of semi-implicit distributions, classical SIVI often resorts to surrogates of evidence lower bound (ELBO) that would introduce biases for training. A recent advancement in SIVI, named SIVI-SM, utilizes an alternative score matching objective made tractable via a minimax formulation, albeit requiring an additional lower-level optimization. In this paper, we propose kernel SIVI (KSIVI), a variant of SIVI-SM that eliminates the need for the lower-level optimization through kernel tricks. Specifically, we show that when optimizing over a reproducing kernel Hilbert space (RKHS), the lower-level problem has an explicit solution. This way, the upper-level objective becomes the kernel Stein discrepancy (KSD), which is readily computable for stochastic gradient descent due to the hierarchical structure of semi-implicit variational distributions. An upper bound for the variance of the Monte Carlo gradient estimators of the KSD objective is derived, which allows us to establish novel convergence guarantees of KSIVI. We demonstrate the effectiveness and efficiency of KSIVI on both synthetic distributions and a variety of real data Bayesian inference tasks.

Deep Learning · Attention Mechanisms

Yingyi Chen, Qinghua Tao, Francesco Tonin, Johan Suykens

While the great capability of Transformers significantly boosts prediction accuracy, it could also yield overconfident predictions and require calibrated uncertainty estimation, which can be commonly tackled by Gaussian processes (GPs). Existing works apply GPs with symmetric kernels under variational inference to the attention kernel; however, omitting the fact that attention kernels are in essence asymmetric. Moreover, the complexity of deriving the GP posteriors remains high for large-scale data. In this work, we propose Kernel-Eigen Pair Sparse Variational Gaussian Processes (KEP-SVGP) for building uncertainty-aware self-attention where the asymmetry of attention kernels is tackled by Kernel SVD (KSVD) and a reduced complexity is acquired. Through KEP-SVGP, i) the SVGP pair induced by the two sets of singular vectors from KSVD w.r.t. the attention kernel fully characterizes the asymmetry; ii) using only a small set of adjoint eigenfunctions from KSVD, the derivation of SVGP posteriors can be based on the inversion of a diagonal matrix containing singular values, contributing to a reduction in time complexity; iii) an evidence lower bound is derived so that variational parameters and network weights can be optimized with it. Experiments verify our excellent performances and efficiency on in-distribution, distribution-shift and out-of-distribution benchmarks.

Probabilistic Methods · Variational Inference

Denis Blessing, Xiaogang Jia, Johannes Esslinger, Francisco Vargas, Gerhard Neumann

Monte Carlo methods, Variational Inference, and their combinations play a pivotal role in sampling from intractable probability distributions. However, current studies lack a unified evaluation framework, relying on disparate performance measures and limited method comparisons across diverse tasks, complicating the assessment of progress and hindering the decision-making of practitioners. In response to these challenges, our work introduces a benchmark that evaluates sampling methods using a standardized task suite and a broad range of performance criteria. Moreover, we study existing metrics for quantifying mode collapse and introduce novel metrics for this purpose. Our findings provide insights into strengths and weaknesses of existing sampling methods, serving as a valuable reference for future developments.

Deep Learning · Generative Models and Autoencoders

Grigory Bartosh, Dmitry Vetrov, Christian Andersson Naesseth

Diffusion models have shown remarkable performance on many generative tasks. Despite recent success, most diffusion models are restricted in that they only allow linear transformation of the data distribution. In contrast, broader family of transformations can help train generative distributions more efficiently, simplifying the reverse process and closing the gap between the true negative log-likelihood and the variational approximation. In this paper, we present Neural Diffusion Models (NDMs), a generalization of conventional diffusion models that enables defining and learning time-dependent non-linear transformations of data. We show how to optimise NDMs using a variational bound in a simulation-free setting. Moreover, we derive a time-continuous formulation of NDMs, which allows fast and reliable inference using off-the-shelf numerical ODE and SDE solvers. Finally, we demonstrate the utility of NDMs through experiments on many image generation benchmarks, including MNIST, CIFAR-10, downsampled versions of ImageNet and CelebA-HQ. NDMs outperform conventional diffusion models in terms of likelihood, achieving state-of-the-art results on ImageNet and CelebA-HQ, and produces high-quality samples.

Probabilistic Methods · Gaussian Processes

Matias Altamirano, Francois-Xavier Briol, Jeremias Knoblauch

To enable closed form conditioning, a common assumption in Gaussian process (GP) regression is independent and identically distributed Gaussian observation noise. This strong and simplistic assumption is often violated in practice, which leads to unreliable inferences and uncertainty quantification. Unfortunately, existing methods for robustifying GPs break closed-form conditioning, which makes them less attractive to practitioners and significantly more computationally expensive. In this paper, we demonstrate how to perform provably robust and conjugate Gaussian process (RCGP) regression at virtually no additional cost using generalised Bayesian inference. RCGP is particularly versatile as it enables exact conjugate closed form updates in all settings where standard GPs admit them. To demonstrate its strong empirical performance, we deploy RCGP for problems ranging from Bayesian optimisation to sparse variational Gaussian processes.

Zach Evans, CJ Carr, Josiah Taylor, Scott Hawley, Jordi Pons

Generating long-form 44.1kHz stereo audio from text prompts can be computationally demanding. Further, most previous works do not tackle that music and sound effects naturally vary in their duration. Our research focuses on the efficient generation of long-form, variable-length stereo music and sounds at 44.1kHz using text prompts with a generative model. It is based on latent diffusion, with its latent defined by a fully-convolutional variational autoencoder. The generative model is conditioned on text prompts as well as timing embeddings, allowing for fine control over both the content and length of the generated music and sounds. It is capable of rendering stereo signals of up to 95 sec at 44.1kHz in 8 sec on an A100 GPU. Despite its compute efficiency and fast inference, the proposed model is one of the best in two public text-to-music and -audio benchmarks and, differently from state-of-the-art models, can generate music with structure and stereo sounds.

Probabilistic Methods · Bayesian Models and Methods

Arnab Bhattacharyya, Sutanu Gayen, Kuldeep S. Meel, Dimitrios Myrisiotis, A. Pavan, N. Vinodchandran

In this paper, we establish a novel connection between total variation (TV) distance estimation and probabilistic inference. In particular, we present an efficient, structure-preserving reduction from relative approximation of TV distance to probabilistic inference over directed graphical models. This reduction leads to a fully polynomial randomized approximation scheme (FPRAS) for estimating TV distances between same-structure distributions over any class of Bayes nets for which there is an efficient probabilistic inference algorithm. In particular, it leads to an FPRAS for estimating TV distances between distributions that are defined over a common Bayes net of small treewidth. Prior to this work, such approximation schemes only existed for estimating TV distances between product distributions. Our approach employs a new notion of *partial* couplings of high-dimensional distributions, which might be of independent interest.

Probabilistic Methods

JIAN XU, Delu Zeng, John Paisley

Deep Gaussian processes (DGPs) provide a robust paradigm in Bayesian deep learning. In DGPs, a set of sparse integration locations called inducing points are selected to approximate the posterior distribution of the model. This is done to reduce computational complexity and improve model efficiency. However, inferring the posterior distribution of inducing points is not straightforward. Traditional variational inference techniques methods to approximate the posterior often leads to significant bias. To address this issue, we propose an alternative named Denoising Diffusion Variational Inference (DDVI) that utilizes a denoising diffusion stochastic differential equation (SDE) for generating posterior samples of inducing variables. We refer to the score matching method in the denoising diffusion model to approximate challenging score functions using a neural network. Furthermore, by combining classical mathematical theory of SDE with the minimization of KL divergence between the approximate and true processes, we propose a novel explicit variational lower bound for the marginal likelihood function of DGP. Through extensive experiments on various datasets and comparisons with baseline methods, we empirically demonstrate the effectiveness of the DDVI method in posterior inference of inducing points for DGP models.

Applications · Neuroscience, Cognitive Science

Umais Zahid, Qinghai Guo, Zafeirios Fountas

We present Langevin Predictive Coding (LPC), a novel algorithm for deep generative model learning that builds upon the predictive coding framework of computational neuroscience. By injecting Gaussian noise into the predictive coding inference procedure and incorporating an encoder network initialization, we reframe the approach as an amortized Langevin sampling method for optimizing a tight variational lower bound. To increase robustness to sampling step size, we present a lightweight preconditioning technique inspired by Riemannian Langevin methods and adaptive SGD. We compare LPC against VAEs by training generative models on benchmark datasets; our experiments demonstrate superior sample quality and faster convergence for LPC in a fraction of SGD training iterations, while matching or exceeding VAE performance across key metrics like FID, diversity and coverage.

Social Aspects · Trustworthy Machine Learning

Darya Biparva, Donatello Materassi

Both the marginal contributions needed for the computation of Shapley values and the graph produced by Pearl-Verma theorem rely on the choice of an ordering of the variables. For Shapley values, the marginal contributions are averaged over all orderings, while in causal inference methods, the typical approach is to select orderings producing a graph with a minimal number of edges. We reconcile both approaches by reinterpreting them from a maximum entropy perspective. Namely, Shapley values assume no prior knowledge about the orderings and treat them as equally likely, while causal inference approaches apply Occam's razor and consider only orderings producing the simplest explanatory graphs. We find that the blind application of Occam's razor to Shapley values does not produce fully satisfactory explanations. Hence, we propose two variations of Shapley values based on entropy maximization to appropriately incorporate prior information about the model. Hence, we propose a variation of Shapley values based on entropy maximization to appropriately incorporate prior information about the model.

Deep Learning · Everything Else

Hyunsu Kim, Ye Gon Kim, Hongseok Yang, Juho Lee

Group Equivariant CNNs (G-CNNs) have shown promising efficacy in various tasks, owing to their ability to capture hierarchical features in an equivariant manner. However, their equivariance is fixed to the symmetry of the whole group, limiting adaptability to diverse partial symmetries in real-world datasets, such as limited rotation symmetry of handwritten digit images and limited color-shift symmetry of flower images. Recent efforts address this limitation, one example being Partial G-CNN which restricts the output group space of convolution layers to break full equivariance. However, such an approach still fails to adjust equivariance levels across data. In this paper, we propose a novel approach, Variational Partial G-CNN (VP G-CNN), to capture varying levels of partial equivariance specific to each data instance. VP G-CNN redesigns the distribution of the output group elements to be conditioned on input data, leveraging variational inference to avoid overfitting. This enables the model to adjust its equivariance levels according to the needs of individual data points. Additionally, we address training instability inherent in discrete group equivariance models by redesigning the reparametrizable distribution. We demonstrate the effectiveness of VP G-CNN on both toy and real-world datasets, including MNIST67-180, CIFAR10, ColorMNIST, and Flowers102. Our results show robust performance, even in uncertainty metrics.

Probabilistic Methods

JIAN XU, Delu Zeng, John Paisley

Deep Gaussian processes (DGPs) provide a robust paradigm in Bayesian deep learning. In DGPs, a set of sparse integration locations called inducing points are selected to approximate the posterior distribution of the model. This is done to reduce computational complexity and improve model efficiency. However, inferring the posterior distribution of inducing points is not straightforward. Traditional variational inference techniques methods to approximate the posterior often leads to significant bias. To address this issue, we propose an alternative named Denoising Diffusion Variational Inference (DDVI) that utilizes a denoising diffusion stochastic differential equation (SDE) for generating posterior samples of inducing variables. We refer to the score matching method in the denoising diffusion model to approximate challenging score functions using a neural network. Furthermore, by combining classical mathematical theory of SDE with the minimization of KL divergence between the approximate and true processes, we propose a novel explicit variational lower bound for the marginal likelihood function of DGP. Through extensive experiments on various datasets and comparisons with baseline methods, we empirically demonstrate the effectiveness of the DDVI method in posterior inference of inducing points for DGP models.